Configuring the looks of memoir's \paragraph

\setparaheadstyle{\normalsize\itshape}
\setafterparaskip{1ex}

Choose the value that suits you; if you want no vertical space use 1sp instead of 0pt as a positive value (even the smallest one) forces a line break.

You probably already know that to number paragraphs you have to say

\setsecnumdepth{paragraph}

which corresponds to the traditional LaTeX

\setcounter{secnumdepth}{4}

If you don't want an indented paragraph just after the title, say also

\setbeforeparaskip{-\beforeparaskip}

which suppresses the indentation, keeping the same vertical spacing before the title.

For what is worth, you must indicate that you're expecting a value, so you have to write \renewcommand*\paragraph[1]{re-definition} this [1] means that you're expecting a value, which you call with #1.

See the wikibook on LaTeX for more details.