CSS Transition font-size: avoid jittering / wiggling

On hover put transition: border-color 0s, font-size 0.3s ease-out;

Because on hover transition: border-color 0s will give only border-color transition not give to font-size.

body {
  font-family: 'Segoe UI', sans-serif;
}
.website {
  width: 180px;
  height: 73px;
  text-align: center;
  line-height: 80px;
  margin: 1px;
  color: white;
  border-bottom: 5px solid darkslateblue;
  background-color: darkslateblue;
  white-space: nowrap;
  overflow: hidden;
  transition: border-color 0.66s ease-out, font-size 0.3s ease-out;
}
.website:hover {
  font-size: 16pt;
  cursor: pointer;
  transition: border-color 0s, font-size 0.3s ease-out;
  border-color: black;
}

<div class="website">Blog 1</div>
<div class="website">Blog 2</div>
<div class="website">Blog 3</div>

You can use CSS transform:scale instead for a smoother transition like so:

.website:hover {
  cursor: pointer;
  transition: border-color 0.4s;
  border-color: black;
}
.website div {
  transition: transform 0.3s ease-out;
}
.website:hover div {
  transform: scale(1.5);
  transition: transform 0s;
}

Here is the JSFiddle demo

Also note that I added the texts within a div and the scaling was done on the div so that the whole box is not scaled :)