Great mathematicians born 1850-1920 (ET Bell's book ≲ x ≲ Fields Medalists)

Nothing very scientific, I'm afraid, but I do have my favourites from this period. Of course, Hilbert, but in addition: Élie Cartan, Emmy Noether, Hermann Weyl, Paul Dirac (if one is allowed to consider him a mathematician), Kurt Gödel, André Weil, Israel Gelfand, Laurent Schwartz. And I'm sure that I'm leaving many in the inkwell, so to speak.

A more scientific approach -- I'm fully aware of its pitfalls! -- is that given our propensity for naming things after people, one could draw a list by counting attributions using the Encyclopaedia of Mathematics or the Princeton Companion as corpus.

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Edit

OK, this time $\epsilon$ more scientifically. I took The Princeton Companion to Mathematics (don't ask), converted the PDF to text, analysed the word frequency and isolated the names mentioned among the answers thus far. I then fed the results to Wordle and obtained the following amusing name cloud:

I've also made available a larger PDF version of the image. Needless to say, one has to take this with a grain of salt. It's a first approximation. For example, no effort has been made to distinguish between Cartans, Artins, 'von Neumann' actually denotes the appearances of 'Neumann'. Not to mention the fact that as Greg points out in a comment below, it is heavily biased in favour of those mathematicians after whom we name things. But I take it we are aware that any such lists are bound to have an element of bias and subjectivity.

The St. Andrews site is an invaluable resource. From that list, I picked (usually) at most one great mathematician born in each year from 1860 to 1910:

$\textbf{EDIT: By popular demand, the list now extends from 1849 to 1920.}$

1849: Felix Klein, Ferdinand Georg Frobenius
1850: Sofia Vasilyevna Kovalevskaya
1851: honorable mention: Schottky
1852: William Burnside
1853: honorable mentions: Maschke, Ricci-Curbastro, Schoenflies
1854: Henri Poincare
1856: Emile Picard (honorable mention: Stieltjes)
1857: honorable mention: Bolza
1858: Giuseppe Peano (honorable mention: Goursat)
1859: Adolf Hurwitz (honorable mention: Holder)
1860: Vito Volterra
1861: honorable mention: Hensel
1862: David Hilbert
1864: Hermann Minkowski
1865: Jacques Hadamard (honorable mention: Castelnuovo)
1868: Felix Hausdorff
1869: Elie Cartan
1871: Emile Borel (honorable mentions: Enriques, Steinitz, Zermelo)
1873: honorable mentions: Caratheodory, Levi-Civita, Young
1874: Leonard Dickson
1875: Henri Lebesgue (honorable mentions: Schur, Takagi)
1877: Godfrey Harold Hardy
1878: Max Dehn
1879: honorable mentions: Hahn, Severi
1880: Frigyes Riesz
1881: Luitzen Egbertus Jan Brouwer
1882: Emmy Amalie Noether (honorable mentions: Sierpinski, Wedderburn)
1884: George Birkhoff, Solomon Lefschetz
1885: Hermann Weyl (honorable mention: Littlewood)
1887: Erich Hecke (honorable mentions: Polya, Ramanujan, Skolem)
1888: Louis Joel Mordell (honorable mention: Alexander)
1891: Ivan Matveevich Vinogradov
1892: Stefan Banach
1894: Norbert Wiener
1895: honorable mention: Bergman
1896: Carl Ludwig Siegel (honorable mention: Kuratowski)
1897: honorable mention: Jesse Douglas
1898: Emil Artin, Helmut Hasse (honorable mentions: Kneser, Urysohn)
1899: Oscar Zariski (honorable mentions: Bochner, Krull, Ore)
1900: Antoni Zygmund
1901: Richard Brauer
1902: Alfred Tarski (honorable mention: Hopf)
1903: John von Neumann (hm's: Hodge, Kolmogorov, de Rham, Segre, Stone, van der Waerden)
1904: Henri Cartan (honorable mentions: Hurewicz, Whitehead)
1905: Abraham Adrian Albert
1906: Kurt Godel, Andre Weil (honorable mentions: Dieudonne, Feller, Leray, Zorn)
1907: Lars Ahlfors, Hassler Whitney (honorable mentions: Coxeter, Deuring)
1908: Lev Pontrjagin
1909: Claude Chevalley, Saunders Mac Lane (honorable mentions: Stiefel, Ulam)
1910: Nathan Jacobson (honorable mention: Steenrod)
1911: Shiing-shen Chern (honorable mentions: Birkhoff, Chow, Kakutani, Witt)
1912: Alan Mathison Turing (honorable mentions: Eichler, Zassenhaus)
1913: Samuel Eilenberg, Paul Erdos, Israil Moiseevich Gelfand (dis/honorable mention: Teichmuller)
1914: honorable mentions: Dantzig, Dilworth, Kac
1915: Kunihiko Kodaira (honorable mentions: Hamming, Linnik, Tukey)
1916: Claude Elwood Shannon (honorable mention: Mackey)
1917: Atle Selberg (honorable mentions: Iwasawa, Kaplansky)
1918: Abraham Robinson
1919: honorable mention: Julia Robinson
1920: Alberto Calderon

As found by Gjergji, a list compiled by James Dow Allen (except for a later Fields Medalist tacked on at the end):

(1854-1912) Poincaré
(1862-1943) Hilbert
(1864-1909) Minkowski
(1865-1963) Hadamard
(1875-1941) Lebesgue
(1877-1947) Hardy
(1879-1955) Einstein
(1881-1966) Brouwer
(1882-1935) Noether
(1885-1955) Weyl
(1885-1977) Littlewood
(1887-1920) Ramanujan
(1892-1945) Banach
(1903-1957) von Neumann
(1903-1987) Kolmogorov
(1906-1978) Gödel
(1906-1998) Weil
(1913-1996) Erdős

This is already not bad. Maybe people can offer improved versions of this starting point. (I personally would both add and subtract from this list.)