Philip was asked to be 'a fourth' member of their luncheon-party.
We have the girl (member no. 1), her chaperon (2), and Lawson (3). That makes Philip 'a fourth'.
Examples of usage:
I have no company but what is proper
To sit with the most flagrant whig at supper.
There's not a man among them but must please,
Since they're as like each other as are peas.
Toland and Hare have jointly sent me word,
They'll come; and Kennett thinks to make a third,
Provided he's no other invitation,
From men of greater quality and station.
(Jonathan Swift, 1714)
Poverty and zeal are an upper and a nether millstone. It is dangerous to make a third in that kind of sandwich. (Ambrose Bierce)
To her cousins she became occasionally an acceptable companion. Though unworthy, from inferiority of age and strength, to be their constant associate, their pleasures and schemes were sometimes of a nature to make a third very useful, especially when that third was of an obliging, yielding temper; and they could not but own, when their aunt inquired into her faults, or their brother Edmund urged her claims to their kindness, that "Fanny was good-natured enough." (Jane Austen)
P.S. The correct form of my answering sentence is probably
Philip was asked to become the fourth member of their luncheon-party.
I retained a to make it more understandable, more corresponing to the original quote.
P.P.S. As an aside, this construction was easy for me to understand, because in Russia we have a much-hackneyed stock question "третьим будешь?" which translates literally as "will you be a third?". It's not that different from "will you make a third?". The question is stereotypically used by (and ascribed to) tipplers who deem it ungentlemanlike to consume beverages in pairs, and so endeavor to gain a quorum of three. The phrase is used in movie episodes, especially of comic nature, and there are a number of jokes, like "Will you be a third? - Yes, sure! -Then let's go and find a second!"