The quid pro quos of going to "MySQLI" include changing your connection string and the query syntax. The one big "gotcha" to me when converting old to new (MySQL to MySQLI) was the lack of a mysqli_result() function in the older days, but now you can do something like this:
$data = $result->fetch_row()[0];
... which seems fairly handy to me in place of mysql_result. I'm not sure when this syntax became valid, off the top of my head.
PDO is generally touted as great for security and portability. I'm not totally convinced, but the use of prepared statements is preferable to its alternative.
I had written quite a bit more here, but I think at this point a good read or two would be in order. I'm not sure there's any 'one size fits all' answer to this issue. This is a thoughtful and well-informed take on the subject. The Reddit comments on it are also kind of interesting.
SitePoint discussion.
Jim Westergren tested, and prefers, PDO with ATTR_EMULATE_PREPARES.
Quora seems to prefer PDO, but there's some really bad info on this page also.
I've been in your shoes, and our management wanted quick-n-dirty. I modified everything to MySQLI with a few search/replace operations in the IDE, adapting the connection code, and replacing calls to mysqli_result(). Took very little time at all.
If they preferred PDO for security and portability, I would've done that. though.